Logarithmic differentiation

Answer Key - Grade 12 Math Worksheet

Circle the best answer, or write your answer on the line.

1. 1.

   Ari uses logarithmic differentiation on y = x^(sin x). Which is y′?

   • y′ = x^(sin x) cos x ln x
   • y′ = x^(sin x)(cos x ln x + (sin x)/x)
   • y′ = cos x ln x + (sin x)/x
   • y′ = sin x · x^(sin x − 1)
2. 2.

   Bo uses logarithmic differentiation on y = x^(eˣ). Which is y′?

   • y′ = eˣ ln x + eˣ/x
   • y′ = eˣ · x^(eˣ − 1)
   • y′ = x^(eˣ) eˣ ln x
   • y′ = x^(eˣ)(eˣ ln x + eˣ/x)
3. 3.

   Esme uses logarithmic differentiation on y = (x² + 1)ˣ. Which is y′?

   • y′ = (x² + 1)ˣ(ln(x² + 1) + 2x²/(x² + 1))
   • y′ = 2x² (x² + 1)^(x − 1)
   • y′ = (x² + 1)ˣ ln(x² + 1)
   • y′ = ln(x² + 1) + 2x²/(x² + 1)
4. 4.

   Jun uses logarithmic differentiation on y = x^(√x). Which is y′?

   • y′ = ln x/(2√x) + 1/√x
   • y′ = x^(√x)(ln x/(2√x) + 1/√x)
   • y′ = x^(√x) ln x/(2√x)
   • y′ = √x · x^(√x − 1)
5. 5.

   Kira uses logarithmic differentiation on y = (x + 1)³/(x + 2)². What is y′ at x = 1?

   20/27
6. 6.

   Luz uses logarithmic differentiation on y = (x + 3)²/(x + 1)². What is y′ at x = 1?

   -2
7. 7.

   Noah uses logarithmic differentiation on y = (x + 2)³/(x + 1)³. What is y′ at x = 0?

   -12
8. 8.

   Pia uses logarithmic differentiation on y = x²√(x + 5). What is y′ at x = 4?

   80/3
9. 9.

   Ren uses logarithmic differentiation on y = x²√(x − 1). What is y′ at x = 2?

   6
10. 10.

    Zee uses logarithmic differentiation on y = x²√(x + 7). What is y′ at x = 2?

    38/3